An Experiment Involving Pendulum and Simple Harmonic Motion - Lab Report Example

Experiment: Pendulum and Simple Harmonic Motion Name: Student number: Professor: Course and Module: Date of performing experiment: Date of submitting report: Abstract Theories covering on acceleration due to gravity can be confirmed by performing an experiment using a pendulum. When a freely suspended pendulum is displaced, it will always experience a force that moves in opposite direction consequently restoring the pendulum to its original equilibrium position. Where the angle of displacement is small, the pendulum is said to be in simple harmonic motion. By varying the length and obtaining the time for each oscillation, a graph of  is plotted. The gradient of this graph is the acceleration due to gravity, which is compared with theoretical value of 9.8m/s2. Introduction This paper is a report covering an experiment that was done on pendulum and simple harmonic motion. The experiment is important in comparing the theoretical acceleration due to gravity given by 9.8m/s2 with the experimental value. The experiment therefore had an objective of determining acceleration due to gravity by using results to conduct an analysis. Mathematical and graphical relationship was indispensible in the analysis. Through the experiment the relationship between period and length surfaced. Theory When a suspended object is displaced, it moves forward and backwards, a movement called oscillation. The timed back and forth movement is commonly referred to as periodic motion (Jearl, 2011). A pendulum that is displaced will always experience a force that moves in opposite direction consequently restoring the pendulum to its original equilibrium position. This means that motion of a displaced pendulum is directed back to equilibrium by a force:  …………….. Eq. 1 The pendulum is therefore said to be in simple harmonic motion. This force, which restores the pendulum to equilibrium position, depends on gravitational force and mass of the pendulum ball: .......................Eq. 2 The period of pendulum is obtained by using: …………..Eq. 3 To be in simple harmonic motion, the pendulum should be displaced by a small angle, in radians (Taylor, 2005). This small angle is given by . Pendulum with and angle of displacement Substituting  for, the force that restores the displaced pendulum to equilibrium becomes:  ……………. Eq. 4 From equation 1 and 4:  And , It is therefore obvious that: …………Eq. 5 This value of K is then substituted to Equation 3 which translates to the period of pendulum given by:  ………………Eq. 6 This is the same as:  ………Eq. 7 This demonstrates that period depends on length, L, and gravitational force of gravity, g. By squaring both sides of the equation, the result is …………….Eq. 8 This is an equation of a line  where; This theoretical background is applicable in analysis to obtain experimental value of acceleration due to gravity. Materials and Method Before setting the experiment, necessary materials were gathered. These materials included: timer, small spherical ball, long and light string, meter stick, support equipment, and weighing scale. The first step entailed weighing the pendulum bob and recording the result. Thereafter, the apparatus was set up by attaching a string to a spherical ball then suspended using a clamp. To ensure that the string pivoted about the same point, the string was clamped between two flat surfaces. Length of the string was then adjusted to 1 meter. This length was obtained by measuring the distance between the point of suspension and the centre of pendulum ball. The reading was tabulated on a table. This step was followed by displacing the suspended pendulum ball by about 50 then released. Time taken for 7 complete oscillations was recorded. The length of the string was then adjusted to 0.9 meters then displaced by about 50. Time taken for 7 complete oscillations was recorded. This procedure was repeated by adjusting the length of the string to 0.8m, 0.7m, 0.6m, 0.5m, and 0.4m then displacing the pendulum balls to 50 and recording time taken for 7 complete oscillations. Data that had been gathered was later used to calculate period of oscillation for individual length starting with 1m, 0.9m, and so on. Period for each length is given by total time divided by number of oscillation i.e. 7 oscillations. Values obtained were entered in a table. Results and Discussion The results following an experiment on a pendulum ball weighing 145.5 g were tabulated in table 1 below. Table 1 L(m) Square root of L(m^1/2) Total time Number of full swings Period of full oscillations (s) T^2 1 1.000 14.41 7 2.059 4.238 0.9 0.949 13.72 7 1.960 3.842 0.8 0.894 12.66 7 1.809 3.271 0.7 0.837 12.2 7 1.743 3.038 0.6 0.775 11.49 7 1.641 2.694 0.5 0.707 10.21 7 1.459 2.127 0.4 0.632 9.06 7 1.294 1.675 From the table above,  Graph of time against square root of length is plotted below. This graph shows that time period increase with an increasing length. Using equation,, the graph below is developed. (N/B: ) The equation of this graph: From this equation, From this experiment, acceleration due to gravity is 9.456 m/s2. This can be compared with the accepted acceleration of 9.8m/s2 as follows: Apparently, difference could result from timing errors. Measurement and estimation errors might also give rise to the difference. Table 2 Results for pendulum ball weighing 45.5 g. L(m) Square root of L(m^1/2) Total time Number of full swings Period of full oscillations (s) T^2 1 1.000 14.31 7 2.044 4.178 0.9 0.949 13.47 7 1.924 3.702 0.8 0.894 12.8 7 1.828 3.342 0.7 0.837 11.97 7 1.71 2.924 0.6 0.775 10.95 7 1.564 2.446 0.5 0.707 10.14 7 1.455 2.117 0.4 0.632 9.18 7 1.33 1.769 Using the data above, graph of time against square root of length is plotted as follows. This graph demonstrates that time period increase with an increasing length. Using equation,, the graph below is developed. The equation of this graph: From this equation, From this experiment, acceleration due to gravity is 9.791 m/s2. This can be compared with the accepted acceleration of 9.8m/s2 as follows: The difference could result from timing errors. Besides, measurement errors might give rise to the difference. Conclusion Using a pendulum weighing 145.5g, acceleration due to gravity was. On the other hand, a pendulum weighing 45.5g gave acceleration due to gravity of. The result shows that slight disparity between actual theoretical value and experimental value. As noted in the discussion, the discrepancy could result from timing and measurement errors. Estimations done during calculation further account for the discrepancies. References Jearl, W. (2011). Principles of physics. 9th Edn. Hoboken, N.J.: Wiley. Taylor, J. R., 2005. Classical Mechanics. Sausalito, CA: University Science Books. Read More